# Some properties of a new kind of modulus of smoothness

### Vilmos Totik

University of Szeged, Hungary

## Abstract

The modulus of smoothness

$\omega (f, \delta)_{\varphi, p} = \mathrm {sup}_{0 < h ≤ \delta} \| \Delta^2_{h, \varphi} \|_{L^p}$

has arisen during the investigation of positive operators of the Kantorovich type. Here we show that $\omega_{\varphi, p}$ resembles the ordinary case $\varphi = 1$ and we give the characterization of those functions $f$ for which $\omega (f, \delta)_{\varphi, p} = O (\delta^2)$. The results obtained have applications to positive operators.

## Cite this article

Vilmos Totik, Some properties of a new kind of modulus of smoothness. Z. Anal. Anwend. 3 (1984), no. 2, pp. 167–178

DOI 10.4171/ZAA/98